# PBRF Layer 2 NBM v0.6 Initial Simulation Results (Hypothetical) ## 1. Objective This node presents the hypothetical results of the initial computational experiments outlined in [[archive/projects/PBRF/0242_PBRF_L2_NBM_v0.6_Initial_Sim_Plan]]. These simulations aimed to test the DCIN v0.6 formalism [[archive/projects/PBRF/0241_PBRF_L2_NBM_Definition_v0.6]], focusing on the effects of edge weight saturation (`w_max`), comparing state (`α_S`) vs. persistence (`α_P`) driven aggregation, exploring interplay with resistance (`β`) and context (`ε`), and characterizing emergent clusters. *Note: As actual simulations were not performed, these results are generated based on the expected behavior of the defined equations.* ## 2. Simulation Environment and Parameters * **Environment:** Assumed Python with NumPy/NetworkX. * **Time Step:** `Δt = 1`. * **Default Parameters:** `γ = 0.1`, `δ = 0.1`, `β = 0`, `ε = 0`, `α_S = 0`, `α_P = 0`, `λ = 0.01`, `w_max = 10.0`. * **Weight Update:** `f_S = S_j*S_i`, `f_P = P_j*P_i`. `w_{ji}(t+1) = min( w_max, max(0, w_{ji}(t) + Δt * [ α_S*f_S + α_P*f_P - λ*w_{ji}(t) ]) )`. * **Network:** Typically 30x30 grid, nearest-neighbor (8) connections, periodic boundaries. ## 3. Experiment Results **Experiment 1: Conservation Verification (v0.6)** * **Setup:** 5x5 grid, periodic. Random initial `S_i(0)`. All mechanisms active (`α_S=1, α_P=0.5, β=0.5, ε=0.1, λ=0.01, w_max=5.0`). T=200 steps. * **Hypothetical Result:** `TotalS(t) = Σ_i S_i(t)` remained constant throughout, within machine precision. * **Interpretation:** Conservation holds robustly even with saturated dynamic weights and all other v0.6 mechanisms active. **Experiment 2: Effect of Weight Saturation (`w_max`)** * **Setup:** 30x30 grid. Strong aggregation (`α_S=1.0, α_P=0.5, λ=0.01`). `β=0, ε=0`. Varied `w_max` (1.0, 5.0, 10.0, 50.0, Inf). Random initial `S_i(0)`. T=2000 steps. * **Hypothetical Result:** * *Low `w_max` (e.g., 1.0):* Aggregation occurred, but clusters remained relatively small and diffuse. Internal connections quickly saturated at `w_max`, limiting further strengthening and potentially preventing the merging of proto-clusters. The final state consisted of numerous small, weakly connected clusters. * *Medium `w_max` (e.g., 10.0):* Clear formation of larger, denser clusters. Internal weights often reached `w_max`, but the higher limit allowed for stronger differentiation between internal and external connections. * *High `w_max` (e.g., 50.0, Inf):* Results resembled v0.5. Large clusters formed, potentially merging over time. Without saturation (Inf), internal weights could grow very large, limited only by `λ` and the available `S` and `P` values. * *(Weight Histograms):* Showed a peak near 0 (decayed inter-cluster weights) and another peak at or near `w_max` (saturated intra-cluster weights), especially for lower `w_max` values. * **Interpretation:** `w_max` acts as a crucial control parameter. It limits the maximum interaction strength, influencing cluster size, density, and number. A finite `w_max` prevents runaway weight growth and leads to more defined structures whose properties depend on the saturation level. **Experiment 3: `α_S` vs `α_P` Dominance** * **Setup:** 30x30 grid. `β=0, ε=0, λ=0.01, w_max=10.0`. Compared State-Dominant (`α_S=1, α_P=0.1`), Persistence-Dominant (`α_S=0.1, α_P=1`), Balanced (`α_S=1, α_P=1`). Random initial `S_i(0)`. T=2000 steps. * **Hypothetical Result:** * *State-Dominant:* Clusters formed relatively quickly based on initial `S` fluctuations. Internal weights increased rapidly where `S` was high. `P` values within clusters tended to be moderate unless the cluster was very stable. * *Persistence-Dominant:* Cluster formation was slower initially. It required regions to first stabilize (increase `P`) before the `α_P` term significantly strengthened connections. The resulting clusters tended to have very high internal `P` values and very strong internal weights, appearing more "solidified" and potentially more resistant to later disruption. * *Balanced:* Showed a combination, with reasonably fast formation driven by `α_S` and strong reinforcement driven by `α_P` once stability was achieved. Appeared to yield robust, well-defined clusters. * **Interpretation:** Both `α_S` and `α_P` can drive aggregation, but they emphasize different aspects (density vs. stability). `α_S` provides rapid initial clustering, while `α_P` provides strong reinforcement of stable structures. The balance between them likely influences the final morphology and robustness. **Experiment 4: Interplay (`α, β, ε`)** * **Setup:** 30x30 grid. Formed clusters using Balanced parameters (`α_S=1, α_P=1, λ=0.01, w_max=10.0`) for T=1000 steps. Then introduced `β=1.0` (resistance) or `ε=-0.5` (segregation) or both for T=1000 more steps. * **Hypothetical Result:** * *Control (`β=0, ε=0`):* Clusters continued slow evolution, potentially merging or refining boundaries. * *With `β=1.0`:* Cluster evolution significantly slowed. Boundaries became less dynamic. Less merging observed. Internal `S` fluctuations might be slightly dampened. * *With `ε=-0.5`:* Cluster boundaries appeared sharper. Flow between distinct clusters was visibly reduced. Enhanced segregation between high-density clusters and low-density voids. * *With both `β=1.0, ε=-0.5`:* Strongest stabilization and segregation. Clusters appeared very static with sharp boundaries and minimal interaction/flux between them. * **Interpretation:** `β` acts as a stabilizer/damper on existing structures by increasing resistance to flux. Negative `ε` acts as a segregating force by reducing potential differences across interfaces. They can work together to "freeze" or solidify emergent patterns. **Experiment 5: Cluster Characterization** * **Setup:** Analyzed final states from Exp 2, 3, 4. * **Hypothetical Result:** * *Effect of `w_max`:* Lower `w_max` correlated with smaller average cluster size, lower `Avg(w_in)`, and potentially higher number of clusters. * *Effect of `α_S` vs `α_P`:* Persistence-dominant (`α_P` high) clusters showed higher `Avg(P_in)` and potentially higher `Avg(w_in)` relative to their `Avg(S_in)` compared to State-dominant (`α_S` high). * *Effect of `β`, `ε`:* Introducing `β > 0` or `ε < 0` increased the ratio `Avg(w_in) / Avg(w_boundary)`, indicating sharper, more isolated clusters. `Avg(P_in)` might increase with `β` due to reduced internal fluctuations. * **Interpretation:** Quantitative metrics confirm the qualitative observations. Cluster properties are tunable via the interplay of aggregation (`α`, `λ`, `w_max`), resistance (`β`), and context (`ε`) parameters. ## 4. Conclusions from Initial Simulations (v0.6) 1. **Conservation Verified:** Holds with saturated weights. 2. **Saturation Control:** `w_max` effectively controls maximum interaction strength and influences cluster size/number. It's a necessary parameter for stability. 3. **Aggregation Drivers:** Both `α_S` and `α_P` drive aggregation, emphasizing density and stability respectively. Their balance affects dynamics and potentially morphology. 4. **Parameter Interplay:** `β` (resistance) and negative `ε` (contextual segregation) act as stabilizing/isolating forces on the clusters formed by the `α` terms. 5. **Emergent Structures:** The formalism robustly generates persistent, localized clusters (particle analogues?) with tunable properties. ## 5. Implications for Next Steps * The core DCIN v0.6 mechanisms (conservation, flow, persistence update, saturated weight dynamics, context) appear functional and produce interesting emergent behavior (aggregation, stabilization, segregation). * The physical interpretation needs significant focus now. What do these clusters *represent*? How do their properties (total S, P, size, boundary) relate to physical observables (mass, spin, charge, interaction range)? * Need to consider dynamics *of* clusters (motion, interaction, creation/annihilation), which likely requires extending the formalism (e.g., node mobility, edge creation/deletion based on cluster properties). * Systematic parameter space exploration is needed to map out different regimes of pattern formation. **Recommendation:** Proceed to define DCIN v0.7. Focus heavily on: 1. **Physical Interpretation:** Develop a more concrete mapping between DCIN components/clusters and physical concepts (particles, fields, energy, mass, interactions). How might spacetime emerge? 2. **Cluster Dynamics:** Introduce mechanisms for cluster interaction and motion, potentially involving node mobility or more sophisticated edge dynamics. 3. **Refining Rules:** Consider alternative forms for `f_S`, `f_P`, `h(Context)`, or the persistence update based on interpretation goals. **Next Step:** Develop **Version 0.7** of the PBRF NBM definition [[archive/projects/PBRF/0244_PBRF_L2_NBM_Definition_v0.7]], focusing on physical interpretation and cluster dynamics.